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Endomorphism rings of reductions of elliptic curves and abelian varieties


Author: Yu. G. Zarhin
Original publication: Algebra i Analiz, tom 29 (2017), nomer 1.
Journal: St. Petersburg Math. J. 29 (2018), 81-106
MSC (2010): Primary 11G10, 14G25
DOI: https://doi.org/10.1090/spmj/1483
Published electronically: December 27, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ E$ be an elliptic curve without CM that is defined over a number field $ K$. For all but finitely many non-Archimedean places $ v$ of $ K$ there is a reduction $ E(v)$ of $ E$ at $ v$ that is an elliptic curve over the residue field $ k(v)$ at $ v$. The set of $ v$'s with ordinary $ E(v)$ has density 1 (Serre). For such $ v$ the endomorphism ring $ \mathrm {End}(E(v))$ of $ E(v)$ is an order in an imaginary quadratic field.

We prove that for any pair of relatively prime positive integers $ N$ and $ M$ there are infinitely many non-Archimedean places $ v$ of $ K$ such that the discriminant $ {\bf\Delta (v)}$ of $ \mathrm {End}(E(v))$ is divisible by $ N$ and the ratio $ \frac {{\bf\Delta (v)}}{N}$ is relatively prime to $ NM$. We also discuss similar questions for reductions of Abelian varieties.

The subject of this paper was inspired by an exercise in Serre's ``Abelian $ \ell $-adic representations and elliptic curves'' and questions of Mihran Papikian and Alina Cojocaru.


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Additional Information

Yu. G. Zarhin
Affiliation: Department of Mathematics Pennsylvania State University University Park, PA 16802 USA
Email: zarhin@math.psu.edu

DOI: https://doi.org/10.1090/spmj/1483
Keywords: Absolute Galois group, Abelian variety, general linear group, Tate module, Frobenius element
Received by editor(s): February 10, 2016
Published electronically: December 27, 2017
Additional Notes: This work was partially supported by a grant from the Simons Foundation (#246625 to Yuri Zarkhin)
Dedicated: Dedicated to Yu. D. Burago on the occasion of his 80th birthday
Article copyright: © Copyright 2017 American Mathematical Society

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